# Geoffrey Colin Shephard

### Quick Info

Manchester, England

Norwich, England

**Geoffrey Shephard**was an outstanding geometer who wrote many interesting and important papers. His books, especially one co-authored with Branko Grünbaum on tilings, have become classics.

### Biography

**Geoffrey Shephard**was the son of William Joseph Shephard (born about 1892) and Nellie Simmonds (born about 1891). William Shephard was a bank clerk who married Nellie Simmonds on 17 April 1919 in the Parish Church of Birch in Rusholme, Manchester. Geoffrey Shephard, the subject of this biography, had one older brother. In 1939, when Geoffrey was twelve years old, the family were living at 41 Carisbrooke Road, Leicester and Geoffrey was attending school in Leicester. His father's occupation at this time is given as bank cashier. In Leicester, Shephard attended Wyggeston Grammar School for Boys. This school, founded in 1876, had Thomas Kingdom (1881-1957) as headmaster when Shephard studied there. Kingdom had studied classics, in particular Greek, at King's College, Cambridge and had been appointed as headmaster at Wyggeston Grammar School in 1920. Shephard graduated from Wyggeston Grammar School in 1945 and, later that year, began studying mathematics at Queens' College, Cambridge.

After a First in Part II of the Mathematical Tripos in 1947 and a special credit in Part III of the Mathematical Tripos in 1948, Shephard remained at Queens' College to study for a Ph.D. under J A Todd. While he was still an undergraduate he wrote an article on reading the sundial which was at Queens' College, originally erected in 1642. The article was published in the Easter 1948 edition of

*The Dial*magazine, dedicated to celebrating the quincentenary of the college. In June 1948 the article was republished as an offprint and made available at the Porters' Lodge for visitors to purchase.

On 11 August 1950 he submitted the paper

*Regular Complex Polytopes*to the London Mathematical Society; it was published in the Proceedings in 1952. The following extract is from the Introduction to the paper:-

Regular polytopes of two or more dimensions in Euclidean space have been extensively studied. The regular polygons and polyhedra, or 'Platonic Solids', have been known since ancient times and the polytopes of four and more dimensions were discovered nearly a century ago by Schläfli. These figures commend themselves to our attention not only on account of their aesthetic appeal but also because their symmetries form important groups isomorphic to groups of real orthogonal matrices.In 1951, Shephard was appointed to a Lectureship in Mathematics at the University of Birmingham but continued to undertake research advised by Todd. His next paper was

The real orthogonal matrix may be considered as a special case of the more general unitary matrix with complex coefficients which satisfies the relation $\bar{T}' = T^{-1}$ where the dash denotes transposition and the bar the complex conjugate. Many groups of unitary matrices are known and it is natural to inquire whether these can be exhibited as symmetry groups of 'complex polytopes' constructed in 'unitary space' ...

Unitary groups are of particular importance since it can be shown that all finite collineation groups can be reduced to this form.

The objects of this paper are:To introduce, for the first time, the concept of a complex polytope.I am very much indebted to Dr J A Todd for his help and advice in the preparation of this paper.

To develop the theory of complex polytopes as far as a complete enumeration of the regular polytopes.

*Unitary Groups generated by reflections*submitted to the

*Canadian Mathematical Journal*on 28 December 1951 and revised on 19 January 1953. In this paper he writes:-

I must express my indebtedness to J A Todd and H S M Coxeter for their advice and suggestions in carrying out the investigations described in this paper. I am especially grateful to the former for undertaking the formidable task of checking the abstract definitions.He was awarded a Ph.D. by the University of Cambridge in 1954 for his dissertation

*Regular Complex Polytopes*. The Abstract of the thesis begins:-

The dissertation consists of a description of a novel method of investigating certain well-known collineation groups in projective space, by the use of 'complex polytopes'. Real polytopes, the analogues in space of any number of dimensions of polygons and polyhedra, have been very fully investigated.In the same year, 1954, he published two further papers, the single author paper

*A construction for Wythoffian polytopes*and the paper

*Finite unitary reflection*groups co-authored with J A Todd. The two author paper has around 500 citations on MathSciNet.

At Birmingham he began a highly successful collaboration with Ambrose Rogers. David Larman writes [14]:-

Leaving a Readership at University College London, in 1954 Ambrose Rogers went to Birmingham as Mason Professor of Pure Mathematics. In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened. In particular, with Geoffrey Shephard, he produced sharp bounds for the volume of a difference body, a problem which had been open for 30 years.In 1957 Shephard produced an updated version of his article on the Queens' College, Cambridge sundial and [27]:-

Shephard's pamphlet was sold at the Porters' Lodge and, with the growth of tourism, became widely known as the authority on the dial.See THIS LINK.

On 27 August 1959, Shephard married Helen Gillian Walker, only daughter of Mr and Mrs F C Walker of 123 Ridgacre Road Birmingham. In 1960 the couple were living at Flat 13, Pakenham Road, Birmingham and in 1965 they were at 6 Mimosa Close, Birmingham.

In 1966 he published the book

*Vector Spaces of Finite Dimension*. He writes in the Preface:-

This book is based on a course of lectures on linear algebra given to second-year Honours students in the University of Birmingham.For more information about this book and other publications by Shephard, see THIS LINK.

In 1967 he became Professor of Pure Mathematics at the University of East Anglia and remained in the chair until he retired in 1984. Soon after going to the University of East Anglia, Shephard began a collaboration with Branko Grünbaum and their first joint paper

*Convex polytopes*was published in the

*Bulletin*of the London Mathematical Society in 1969. This major survey paper covered the advances in the subject in the three years following the publication of Grünbaum's book

*Convex polytopes*(1966). We note that MathSciNet lists 65 joint publications by these two mathematicians. Moshe Rosenfeld writes [18]:-

Branko met Geoffrey Shephard in 1975. They decided to write a book on Visual Geometry. This was a huge undertaking. So they decided to start with tilings and patterns as a first step in their program. After 11 years of research, tracing ancient and current places where tilings were used, the book was published.It is unclear what 'met' means in this quote. Certainly they wrote their first joint paper in 1969 so one assumes that 'met' means 'met in person' but that seems to be contradicted by the note in the four author paper

*The enumeration of normal 2-homeohedral tilings*(1986) by Branko Grünbaum, H D Löckenhoff, G C Shephard and Á H Temesvári which reads:-

This paper represents, we believe, an excellent example of international cooperation. The four authors, from four different countries, have never met and do not have a common language. The preparation of the final manuscript and diagrams was undertaken by two of the authors (B G and G C S) who take full responsibility for any errors that may occur.The book that took Shephard and Grünbaum eleven years to produce was

*Tilings and Patterns*(1986). The Publisher quotes from reviews:-

"Remarkable ... It will surely remain the unique reference in this area for many years to come," Roger Penrose, Nature; "... an outstanding achievement in mathematical education," Bulletin of the London Mathematical Society; "I am enormously impressed ... Will be the definitive reference on tiling theory for many decades. Not only does the book bring together older results that have not been brought together before, but it contains a wealth of new material ... I know of no comparable book," Martin Gardner.Joseph Malkevitch writes in the review [16]:-

What Grünbaum and Shephard have done, in a dazzling display of scholarship, erudition, and research, is collect in one volume a compendium of the accumulated knowledge about tilings and patterns developed by a wide range of individuals including artisans and craftsmen, mathematicians, crystallographers, and physicists. In doing so they were forced to take a fresh look at what was known, what was thought to be known, and what had yet to be investigated, and to provide a framework in which all of this information could be succinctly put down. The project, which was started about 10 years ago and has only now been brought to (partial) completion, is well worth the wait.For more extracts from reviews of this book, see THIS LINK.

Shephard had written another book which he had published well before

*Tilings and Patterns*, namely the joint work

*Convex polytopes and the upper bound conjecture*written in collaboration with Peter McMullen (born 11 May 1942). In fact McMullen had been a Ph.D. student at the University of Birmingham advised by Shephard. When Shephard moved to the University of East Anglia in 1967, McMullen went with him. He had been awarded a Ph.D. in 1968 for his thesis

*On the Combinatorial Structure of Convex Polytopes*. In a review of

*Convex polytopes and the upper bound conjecture*, Donald Coxeter writes [2]:-

This carefully composed book summarises and extends some of the most important parts of B Grünbaum's treatise 'Convex polytopes' (1967). It is a more readable exposition, and brings the subject up to date.In his paper

*Space Filling with Identical Symmetrical Solids*(1985), Shephard gives his opinion about the lack of geometry teaching [24]:-

It seems to us that geometry of this kind described here (concerning polyhedra, space-fillings, symmetry groups, etc.) is both interesting and exciting. It is a tragedy that our present educational system completely ignores these subjects and the little time available for teaching geometry seems to be devoted to comparatively simple and uninteresting topics. Perhaps this is one of the reasons why there still remain so many unsolved geometrical problems in the familiar three-dimensional space in which we live.In fact he had a reputation as an outstanding teacher as Arnold Shaw points out in [22]:-

[Geoffrey Shephard was] an inspiring speaker and with a sense of theatre and an ability to hold an audience - at least in the few University of East Anglia lectures of his which I attended. He reminded one of G H Hardy's dictum along the lines of: You may have complicated things to say, but you should always say them simply. It comes as no surprise that he provided a prize for work in mathematics which can be explained to people who have no prior knowledge in that field.The prize that Shaw refers to in this quote is the 'Shephard Prize' awarded by the London Mathematical Society [23]:-

Following a very generous donation made by Professor Geoffrey Shephard, the London Mathematical Society will, in 2015, introduce a new prize. The prize, to be known as the Shephard Prize will be awarded biennially. The award will be made to a mathematician (or mathematicians) based in the UK in recognition of a specific contribution to mathematics with a strong intuitive component which can be explained to those with little or no knowledge of university mathematics, though the work itself may involve more advanced ideas.See THIS LINK.

...

Professor Shephard himself is Professor of Mathematics at the University of East Anglia whose main fields of interest are in convex geometry and tessellations. Professor Shephard is one of the longest-standing members of the London Mathematical Society, having given more than sixty years of membership. The Society wishes to place on record its thanks for his support in the establishment of the new prize.

Shephard retired from his chair at the University of East Anglia in 1984. Following his retirement, he taught as a Professorial Fellow for three years, before being appointed Emeritus Professor in 1987. Retirement, however, did not in any way mean that he stopped undertaking research and writing papers. In fact MathSciNet lists 48 papers written by Shephard (some are joint works) which were published between 1987 and 2016. Shaun Stevens wrote in 2016 [26]:-

He continued to be a regular visitor to the School of Mathematics since that time [1987]; it is only in the last few years that these visits became less frequent.Among the honours he received was a Sc.D. by the University of Birmingham for "a significant contribution to mathematical knowledge."

When Shephard's health deteriorated and he required palliative care, he entered the Priscilla Bacon Lodge in Norwich where he died just a few days before his 89th birthday.

### References (show)

- W H Cockcroft, Review: Vector Spaces of Finite Dimension, by Geoffrey C Shephard,
*The Mathematical Gazette***51**(376) (1967), 177-178. - H S M Coxeter, Review: Convex polytopes and the upper bound conjecture, by Peter McMullen and Geoffrey C Shephard,
*Mathematical Reviews*MR0301635**(46 #791)**. - H S M Coxeter, Review:
*Mathematical Reviews*MR0500451**(58 #18090)**. - H S M Coxeter, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*Mathematical Reviews*MR0857454**(88k:52018)**. - G Ewald, Satins and Twills: An Introduction to the Geometry of Fabrics (1980), by Branko Grünbaum and Geoffrey C Shephard,
*Mathematical Reviews*MR0600071**(82k:52017)**. - R Ding, Pick's Theorem (1993), by Branko Grünbaum and Geoffrey C Shephard,
*Mathematical Reviews*MR1212401**(94j:52012)**. - J Donegan, Review: Tilings and Patterns, An Introduction, by Branko Grünbaum and G C Shephard,
*The Mathematics Teacher***83**(2) (1990), 167. - L Fejes Tóth, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*Bulletin of the American Mathematical Society***17**(1987), 369-372. - P Garcia, Review: Tilings and Patterns, An Introduction, by Branko Grünbaum and G C Shephard,
*The Mathematical Gazette***74**( 468) (1990), 207-209. - Geoffrey Shephard,
*London Mathematical Society Newsletter***461**(September 2016), 26. - Geoffrey Shephard,
*Eastern Daily Press*(24 August 2016). - Geoffrey C Shephard,
*The Record*, Queens' College 2017.

https://www.queens.cam.ac.uk/sites/www.queens.cam.ac.uk/files/publicationFiles/queens_the_record_2017_-_high_res.pdf - S W Golomb, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*The American Mathematical Monthly***95**(1) (1988), 63-64. - D Larman, Ambrose Rogers,
*London Mathematical Society Newsletter***344**(January 2006), 7. - J Malkevitch, In Praise of Collaboration,
*York College, CUNY*(31 March 2021).

https://mathvoices.ams.org/featurecolumn/2021/03/31/fc-2021-04/ - J Malkevitch, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*Science, New Series***236**(4804) (1987), 996-997. - R F Rinehart, Review: Vector Spaces of Finite Dimension, by Geoffrey C Shephard,
*Mathematical Reviews*MR0222096**(36 #5148)**. - M Rosenfeld, Branko Grünbaum: the mathematician who beat the odds,
*The Art of Discrete and Applied Mathematics*.

https://adam-journal.eu/index.php/ADAM/article/download/1520/1303 - W J Satzer, Review: Tilings and Patterns (Dover reprint), by Branko Grünbaum and G C Shephard,
*Mathematical Association of America.*

https://www.maa.org/press/maa-reviews/tilings-and-patterns - R L E Schwarzenberger, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*Bulletin of the London Mathematical Society*20 (1988), 167-192. - M Senechal, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*American Scientist***75**(5) (1987), 521-522. - A Shaw, A Sense of Theatre,
*Eastern Daily Press*(1 April 2021).

https://web.archive.org/web/20210907133901/https://www.familynotices24.co.uk/edp/view/4167811/geoffrey-shephard - Shephard Prize: New Prize for Mathematics,
*London Mathematical Society Newsletter***437**(June 2014), 1. - G C Shephard, Space Filling with Identical Symmetrical Solids,
*The Mathematical Gazette***69**(448) (1985), 117-120. - G C Shephard, My work with Branko Grünbaum,
*Geombinatorics***9**(2) (1999), 42-48. - S Stevens, Prof Geoffrey Shephard, University of East Anglia (2017).

https://web.archive.org/web/20170630023826/https://www.uea.ac.uk/alumni/your-uea/latest-news/prof-geoffrey-shephard - R Walker, History of the Dial,
*Queens' College, Cambridge*(January 2017).

https://www.queens.cam.ac.uk/visiting-the-college/history/college-facts/the-buildings/history-of-the-dial - J A Wenzel, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*The Mathematics Teacher***80**(6) (1987), 497-498. - H C Williams, Review: Tilings and Patterns, by Branko Grünbaum and G C Shephard,
*The Mathematical Gazette***71**(458) (1987), 347-348.

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Written by J J O'Connor and E F Robertson

Last Update February 2023

Last Update February 2023